The optimal electromagnetic carrier frequency balancing structural and metrical information densities with respect to heat removal requirements

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Optics Communications 94 ( 1992 ) 13-18 North-Holland

O P T I C S COMMUNICATIONS

The optimal electromagnetic carrier frequency balancing structural

and metrical information densities with respect to heat removal

requirements

H a l d u n M . O z a k t a s

Department of Electrical Engineering, Bilkent University, 06533 Bilkent, Ankara, Turkey

a n d

J o s e p h W . G o o d m a n

Department of Electrical Engineering, Standford University, Stanford, CA 94305, USA

Received 3 February 1992; revised manuscript received 5 June 1992

The use of higher electromagnetic carrier frequencies for communication in a computing environment results in both increased spatial information density and larger available modulation bandwidth. However, assuming that the communication energies are dissipated, the heat that must be removed from unit volume per unit time increases quickly with higher frequencies, resulting in a maximum useful frequency based on our limited ability to remove heat. We show that this frequency is relatively insensitive to system specific parameters and estimate its order of magnitude to lie near the infrared and visible bands of the spectrum.

1. Introduction

O f the four basic forces in nature, only the elec- t r o m a g n e t i c force is effective on a scale c o m p a r a b l e to biological organisms. T h e h u m a n visual a n d ner- vous systems are essentially b a s e d on electromagnetic interactions. M o s t m a n m a d e c o m p u t i n g sys- t e m s rely on e l e c t r o m a g n e t i c p h e n o m e n a for b o t h re- generative nonlinearities a n d c o m m u n i c a t i o n a m o n g their elements. T h e n o n l i n e a r o p e r a t i o n s m a y be b a s e d on electronic i n t e r a c t i o n .

E l e c t r o m a g n e t i c wave p r o p a g a t i o n is a m o s t basic m e a n s o f i n f o r m a t i o n t r a n s m i s s i o n . In this w o r k we will c o n s i d e r the use o f m o d u l a t e d e l e c t r o - m a g n e t i c c a r r i e r waves to establish c o m m u n i c a t i o n b e t w e e n the e l e m e n t s o f a c o m p u t i n g system. We a s s u m e t h a t the signals are g u i d e d to t h e i r d e s t i n a t i o n s with di- electric m e d i a only ( i n c l u d i n g lenses, holograms, waveguides etc.), the use o f c o n d u c t o r s is e x c l u d e d f r o m c o n s i d e r a t i o n .

The e l e m e n t s o f o u r c o m p u t i n g s y s t e m m a y be relatively s i m p l e switching devices o r relatively c o m -

plex processing elements. In any event we will as- s u m e t h e m to be very small in size. We w o u l d like our overall system to be as c o m p a c t as possible. Being able to handle vast a m o u n t s o f i n f o r m a t i o n in a small v o l u m e is not only a m e r i t in itself, as e x e m p l i f i e d by the h u m a n eye, b u t also results in s m a l l e r speed- of-light l i m i t e d c o m m u n i c a t i o n delays between dis- tant e l e m e n t s o f o u r system. Spatial i n f o r m a t i o n d e n s i t y a n d heat r e m o v a l are two m a j o r physical c o n s i d e r a t i o n s which will l i m i t how densely we can p a c k the elements o f our c o m p u t i n g system. These c o n s i d e r a t i o n s are i n t i m a t e l y tied together through the carrier frequency. Increasing the carrier frequency i m p r o v e s spatial i n f o r m a t i o n density, however also increases the a m o u n t o f heat we m u s t re- m o v e p e r unit cross section a n d time. Since o u r heat r e m o v a l ability is limited, there exists an o p t i m a l carrier frequency resulting in smallest possible sys- t e m size a n d delay.

In the next section we discuss spatial i n f o r m a t i o n d e n s i t y a n d heat removal. In the t h i r d section we in- t r o d u c e a s i m p l e m o d e l c o m p u t i n g system a n d show

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that the optimal carrier wavelength is insensitive to system specific parameters (such as the number of elements, number of connections per element and interconnection topology) and lies near the infrared and visible bands of the spectrum. Finally, we briefly mention the importance of this band of frequencies in other areas of physical science.

Owing to the general nature of this study, our analysis is necessarily approximate. We have also pre- ferred to leave out certain geometrical factors of the order of unity for simplicity and generality.

2. Physical mechanisms

In this section we discuss the two major physical mechanisms which will limit how closely we can pack the elements of our system.

2.1. Spatial information density

We will assume that information transfer takes place along each independent spatial channel in the form of binary digital pulses impressed on a sinu- soidal electromagnetic carrier of wavelength 2. It is a well known property of wave propagation that a cross sectional area of at least ~ 2 2 must be allocated per each independent spatial channel. A more general formulation states that the total volume allocated for communication in a system with total interconnection length ltota~ must at least be [ 1 ]

• ~ 2ltota I . ( 1 )

Thus we see that the spatial (or

structural

[2] ) information density can be increased by reducing the carrier wavelength 2, or in other words by increasing the frequency f

2.2. Energy dissipation and heat removal

The energy of a single photon of electromagnetic radiation is given by

hf=hc/2

where h is Planck's constant and c is the speed of light. With increasing frequencies, the increasing energy of a single photon will require larger energies to maintain reliable communication, leading to a decrease in the so called

metrical

[ 3 ] information density. In general, based on statistical considerations 0>/1 photons will be re-

quired per transmitted bit. The human eye, under optimum conditions, can detect as little as 100 photons per second [ 4 ]. If it is assumed that the eye can still differentiate events spaced about 100 ms apart under these conditions, this corresponds to 10 photons per bit. Properly designed shot-noise limited systems may require ~ 100-1000 photons, depend- ing on the error rate we are willing to tolerate. We will follow Smith [5] in taking O= 103. The actual communication energies involved in practical systems may be quite larger than

E=Ohc/2

due to various forms of overhead.

The energy associated with each switching or regeneration event can usually be decreased with de- creasing temperatures [ 6,7 ]. Thus for sufficiently low temperatures and operating voltages, the communication energies will dominate electronic switching and regeneration energies in the elements, so that the latter can be ignored. We are assuming that these energies are irreversibly dissipated.

Two remarks are appropriate at this point. First, we note that in principle, the energy E associated with each transmitted bit may be reduced down to

E ~ hB,

where B is the bit transmission rate [8,9]. Second, we are assuming that the energy associated with each transmitted bit of information is dissipated. It has been argued by many authors that this is not a fun- damental necessity [ 10-13 ]. Thus, our assumptions do not correspond to ultimate physical limitations, but rather to an idealization of existing dissipative short-noise limited communication systems.

The dissipated energy must be removed from the system. Whatever the modality (conduction, convection or radiation), heat transfer can only take place

through

a surface (flux conservation). This is most easily visualized by considering the flow of a fluid coolant through our system, which is assumed to occupy a cubic volume (fig. 1 ). Thus, as discussed at length in ref. [ 14 ], our ability to remove heat from a three-dimensional system (which for the purpose of this paper is assumed to be cubic in shape) can be characterized by a quantity Q, the amount of power we can remove per unit cross sectional area of the cubic system. (For instance, we would be able to remove

QL 2

of power from a cube with edge length L.) The value of Q is estimated as [ 14 ]

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Volume 94, number 1,2,3 OPTICS COMMUNICATIONS 1 November 1992

/

Fig. 1. Heat removal via fluid convection. The maximum amount o f heat we may remove per unit time is proportional to the cross sectional area o f the system.

where p, Cs, x a n d / t denote the density, heat capac- ity, thermal conductivity and viscosity of the fluid coolant, respectively. AT is the maximum allowed temperature rise and AP is the applied hydraulic dif- ference across the channels of fluid flow. An upper limit

AP~pCsAT/2

is set by the necessity to limit viscous power dissipation. Thus, the maximum value of Q is found as

Q ~ 0.2

(p2C~x/it)1/2AT3/2.

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According to ref. [ 15 ], water fares best in terms of maximizing the above expression. With A T = 100 K, we find Q ~ 106 W / c m 2, a rather optimistic estimate several orders of magnitude beyond what is cur- rently achieved.

Ultimately, regardless of the specifics of our heat removal system, how much we can increase Q is limited by material parameters which are in turn related to atomic constants. This, even allowing for ad- vances in materials; we do not expect that the value of Q can be further improved beyond a few more orders of magnitude, if we are to construct our processing systems from solid state materials under 'earthly' conditions. Whereas such a conclusion is too imprecise to have any engineering value, it will suf- fice for our purpose.

3. Model computing system

Our model computing system is to be constructed by establishing a prespecified pattern of connections among an array of N ~/3 X N ~/3 X N 1/3 elements laid

out on a regular cubic grid with as yet unspecified lattice constant d. The system is confined in a cubic box of volume

Nd 3. Let k

denote the average number of connections per element and ~ denote the average length of the connections in units of grid spacing. Ig- noring numerical factors of the order of unity, we have

1 <~ g<~

N 1/3.

The total connection length in real units is/total =

kNgd,

since there are a total of

kN

connections. Using eq. ( 1 ), we find that a volume of at least

~22kNrd

is necessary to establish these connections. We must choose d so that the system volume

Nd 3

exceeds this minimum required volume plus the volume of the elements. Assuming the volume occupied by the elements to be negligible, we find that d must satisfy

d2>~22k?.

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The total amount of power dissipated is

kNEB

where

E=Ohc/2

and B is the rate at which bits of data are being emitted into each connection. We again refer to fig. 1 and assume that we are able to remove Q of power through unit cross sectional area. Since the cross sectional area of our system is

N2/ad2,

the total amount of power we can remove is

QN2/ad2.

Requiring that this be greater than the total power dissipated, we find

kN 1/ 30hcB

d2> (5)

2Q

We would like to minimized d both for sake of maintaining a compact system and - when it is a lim- iting factor - minimizing speed-of-light limited communication delays. The delay across the extent of the system is given by

z=Nl/3d/c.

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Thus, given N and B, one may choose 2 so as to min- imize d and z. Equating the right hand sides of eqs. (4) and ( 5 ) and solving for 2 we obtain, with O= 103 and Q = 100 W/era 2,

2 =

(ch/Q)1/3191/3B

1/3( 1, N 1/9}

~ 0 . 5 B 1 / 3 { 1 ,

N 1/9}

~tm, ( 7 ) where B is in Gbit/s, and the notation {1,

N 1/9}

means that any value between the two extremes is possible. Most present day computers operate at rates of 10-1000 Mbit/s. Switches operating at ~ 100

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G b i t / s have been built. How large can N be? The connection machine [ 16 ] has ~ 10 5 elements. Some computers may have 10 8 transistors or more. The human brain has 1011 neurons [ 17 ]. Although both B and N may vary over a large range, the optimal value of 2 is quite insensitive to this variation.

In most cases, the rate B at which pulses of information are emitted into each connection will be related to the cycle time of the system, determined by the worst case signal delay z, as given by ~1 eq. (6). This approach was taken by Keyes [6 ]. Thus let us

write B =

fl/r

where fl is a constant. Most authors take

r = 1 without discussion, however there is no reason why fl should not be larger (i.e. we may allow pi- pelining). This time, using eqs. (4), ( 5 ) with equal-

ity, eqs. (6), (7) and B =

fl/z

we might solve for the

optimal value of 2 as

]~1/4

2 =

(c2h/Q)1/401/4

kl/8 {

1, N

1/8}

•1/4

~ 1 5 _{k 1 / 8 { 1 ' N 1 . 8 }}_~tm. ₍₈₎

The optimal wavelength is very insensitive to both how it is calculated and to the various parameters. Equation (8) is plotted in fig. 2 for g= 1 and ?=

N 1/3

with Q as a parameter.

4. Analysis

In this article, we considered the use of a full three- dimensional layout which may not always be possible to realize. Our analysis may be repeated for a fully two-dimensional planar layout, for which the optimal wavelength is found to be of the order of ~ 0.1 p.m. The optimal wavelength is smaller in this case because a fully two-dimensional layout is much more restrictive in terms of providing communication. The optimal wavelength is also somewhat more strongly dependent on the system parameters in two- dimensions. For layouts lying between these two extremes, which combine realizability and flexibility in

~t Here we are implicitly a s s u m i n g that the length o f the longest interconnection is of the order o f the linear extent o f the system. We should also note that z m a y not be o f direct signifi- cance in some cases, such as a nearest neighbor connected system.

providing communication, the optimal wavelength will lie between this value and those found above for full three-dimensional layouts.

Apart from system parameters with limited effect, the optimal carrier wavelength is given essentially by

2 ~ (cZh/Q)1/4

which is equal to a few ~tm for Q = 100 W / c m 2. H u m a n eyes, as well as those of many other living beings, operate at wavelengths around 0.5 ~tm. Even bees, despite severe diffraction problems, have developed compound eyes enabling them to operate at only slightly higher frequencies [ 18 ]. There exist 'windows' at visible frequencies in the absorption spectrum of water [ 19] and the spectral distribution of solar radiation arriving at the earth [20,21 ]. Many electronic energy levels, in particular semiconductor bandgaps, correspond to visible and infrared frequencies, enabling the construction of efficient sources and detectors at these frequencies [22]. Al- though it would be unwarranted for us to draw any conclusions from these facts, one is tempted to think that they are not unrelated. After all, Q is intimately related to atomic constants through material parameters.

5. Conclusion

We considered the use of electromagnetic radiation for communication among the elements of a computing system. In general, increasing the frequency of radiation enables higher information densities which potentially offer smaller system size and communication delays. Even when delay is not an issue, the ability to handle vast amounts of information in a compact volume is desirable in itself. In- creasing the carrier frequencyfalso increases the energy per transmitted bit. Thus, assuming these energies are dissipated, the amount of heat that must be removed from unit volume per unit time quickly increases. The m a x i m u m removable heat pertaining to 'earthly' conditions may be estimated based on physical considerations. After a certain frequency, we can no longer benefit from the high information and packing densities offered because we will fail to meet the heat removal requirements.

We have shown that there exists an o p t i m u m carrier frequency balancing information density and heat removal imposed bounds on element spacing result-

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Volume 94, number 1,2,3 OPTICS COMMUNICATIONS 1 November 1992 E I0 4 103 t0 2 10 j 10 o , , I T J I H f , I , , , , , ± _ _ 0.1 W/cm2 _ _ 10 W/cm2 ... 1 KW/cm2 _._'. 100 KW/cm2 10 1 10_2 f . . . L I0 4 10 5 i i i t J , , , , , , i , , , * , , i I i i i i i J i i i i i J ~ l l ~ i ~ ~ 1 1 1 1 106 107 108 109 1010 N

Fig. 2. The optimal carrier wavelength for full three-dimensional layouts. Equation (8) is plotted with Q as a parameter. We take 0 = 103

andfl~/4/k~/8= 1. The horizontal lines correspond to g= 1 and the slanted lines correspond to ~ = N 1/3.

ing in smallest possible system size. This frequency is found to depend on the speed o f light, which sim- ply relates the frequency to the m i n i m u m resolvable dimension 2, on Planck's constant, which relates the frequency to the m i n i m u m resolvable energy, on our heat removal ability as quantified by Q and also on other system parameters. Since the optimal value o f f i s only weakly dependent on all parameters, it was possible to obtain a system independent estimate o f f ~ 1014 which corresponds to the infrared and visible

bands.

Needless to say, care must be exercised in arriving at any practical conclusions from our results which have been derived for an idealized computing sys- tem limited by simple physical considerations only.

Acknowledgements

The first author acknowledges the support o f the Alexander von Humboldt Foundation through a Post Doctoral Research Fellowship.

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